Vinay and Varma can do a work in 30 days and 60 days respectively. If they work on alternate days beginning with Vinay, in how many days will the work be completed?

Options

A. 45
B. 35
C. 40
D. 50

Explanation

To solve the problem of how many days Vinay and Varma will take to complete the work when they work on alternate days, we can follow these steps:

### Step 1: Determine the work done by each person in one day.
- Vinay can complete the work in 30 days. Therefore, his one day work is:
\[ \text{Vinay's one day work} = \frac{1 \text{ work}}{30 \text{ days}} = \frac{1}{30} \text{ work/day} \]

- Varma can complete the work in 60 days. Therefore, his one day work is:
\[ \text{Varma's one day work} = \frac{1 \text{ work}}{60 \text{ days}} = \frac{1}{60} \text{ work/day} \]

### Step 2: Calculate the total work done in two days.
- Since they work on alternate days starting with Vinay, in two days:
- Day 1: Vinay works and completes \(\frac{1}{30}\) of the work.
- Day 2: Varma works and completes \(\frac{1}{60}\) of the work.

- Total work done in two days:
\[ \text{Total work in 2 days} = \frac{1}{30} + \frac{1}{60} \]

- To add these fractions, find a common denominator (which is 60):
\[ \frac{1}{30} = \frac{2}{60} \]
\[ \text{Total work in 2 days} = \frac{2}{60} + \frac{1}{60} = \frac{3}{60} = \frac{1}{20} \]

### Step 3: Determine how many such two-day cycles are needed to complete the work.
- If \(\frac{1}{20}\) of the work is done in 2 days, then the total work (1 unit) will be completed in:
\[ \text{Number of two-day cycles} = 20 \text{ cycles} \]

- Total days taken for 20 cycles:
\[ \text{Total days} = 20 \times 2 = 40 \text{ days} \]

### Final Answer:
Vinay and Varma will complete the work in **40 days**.

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